×

Necessary and sufficient conditions for the oscillation of forced nonlinear second order differential equations with delayed argument. (English) Zbl 0884.34075

Under the assumption that \(a(t)\) is nonnegative, there exists a bounded solution \(h(t)\) of \[ h''(t)= g(t)\quad\text{and}\quad\phi(t)\leq t,\;\lim_{t\to\infty} \phi(t)=\infty, \] necessary conditions for equation \(x''(t)+ a(t)|x(\phi(t))|^\nu\cdot \text{sgn }x(\phi(t))= g(t)\) to be oscillatory in the sublinear and the superlinear cases are established and when \(h(t)\) has special properties, these conditions are also sufficient.

MSC:

34K11 Oscillation theory of functional-differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Atkinson, F. V., On second order nonlinear oscillation, Pacific J. Math., 5, 643-647 (1955) · Zbl 0065.32001
[2] Belohorec, S., Oscillatory solutions of certain nonlinear differential equations of the second order, Mat.-Fyz. Casopis Sloven. Acad. Vied., 11, 250-255 (1961) · Zbl 0108.09103
[3] El-Sayed, M. A., An oscillation criterion for forced second order linear differential equation, Proc. Amer. Math. Soc., 118, 813-817 (1993) · Zbl 0777.34023
[4] Kartsatos, A. G., On the maintenance of oscillation of \(n\), J. Differential Equations, 10, 355-363 (1971) · Zbl 0211.11902
[5] Kartsatos, A. G., On the maintenance of oscillation under the effect of periodic forcing term, Proc. Amer. Math. Soc., 33, 377-382 (1972) · Zbl 0234.34040
[6] Komkov, V., On boundedness and oscillation of the differential equation \(xAtgxft^n\), SIAM J. Appl. Math., 22, 561-568 (1972) · Zbl 0223.34029
[7] Macki, J. W.; Wong, J. S.W., Oscillation of solutions to second order nonlinear differential equations, Pacific J. Math., 24, 111-117 (1968) · Zbl 0165.42402
[8] Rankin, S. M.W., Oscillation theorems for second order nonhomogeneous linear differential equations, J. Math. Anal. Appl., 53, 550-553 (1976) · Zbl 0328.34033
[9] Wong, J. S., Second order nonlinear forced oscillations, SIAM J. Math. Anal., 19, 667-675 (1988) · Zbl 0655.34023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.